Calculus 1 : How to find solutions to differential equations

Study concepts, example questions & explanations for Calculus 1

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Example Questions

Example Question #351 : Equations

Find  if 

Possible Answers:

Correct answer:

Explanation:

For this problem, note that:

Product Rule 

To solve this problem, differentiate the expression one variable at a time, treating other variables as constants:

If we're looking for  for the function  then we'll begin by differentiating with respect to  first:

Next, differentiate with respect to :

Now finally we'll differentiate with respect to ; remember to use the product rule:

Example Question #101 : How To Find Solutions To Differential Equations

Find  for the function 

Possible Answers:

Correct answer:

Explanation:

For this problem, note that:

Product Rule: 

To solve this problem, differentiate the expression one variable at a time, treating other variables as constants:

To find  for the function , begin by differentiating with respect to :

Next, differentiate with respect to :

Finally, differentiate with respect to  once more, remembering to utilize the product rule:

Example Question #353 : Equations

Find  for the equation

Possible Answers:

Correct answer:

Explanation:

For this problem, note that:

Take the derivative of each term in the equation twice: with respect to  and then with respect to . When taking the derivative with respect to one variable, treat the other variable as a constant.

For the function

The derivative is then

Now bring  and  terms to opposite sides of the equation:

Finally, rearrange terms to find :

Example Question #352 : Equations

Find the derivative of the following function.

Possible Answers:

None of these

Correct answer:

Explanation:

To solve this derivative, we must realize that there are two parts to the function and we must use the product rule. The rule states that  .

We must asle recognize that the derivative of  is  and the derivative of  is . By these rules, the derivative is

Example Question #102 : How To Find Solutions To Differential Equations

Find the derivative of the function.

Possible Answers:

None of these

Correct answer:

Explanation:

This function is just a function inside of a function. This means we have to use the chain rule. The chain rule states that the derivative of .

The derivative of  is  and the derivative of  is . This makes the derivative 

This makes sense because .

Example Question #103 : How To Find Solutions To Differential Equations

Find the derivative of the function.

Possible Answers:

None of these

Correct answer:

Explanation:

To find the derivative of this function, we must use the division rule. This rule states that the derivative of  is . The derivative of  is  and the derivative of  is .

Thus the derivative is 

Example Question #104 : How To Find Solutions To Differential Equations

Find the derivative of the function.

Possible Answers:

None of these

Correct answer:

Explanation:

To find the derivative of this function we need to use the chain rule and multiplication rule. The chain rule states that the derivative of  is . The multiplication rule states that the derivative of  is. The derivative of  is . The derivative of sin is cos and the derivative of cos is -sin. So lets say

 then  and 

 then 

So the answer is

Example Question #362 : Equations

What is the slope of the function  at the point ?

Possible Answers:

Correct answer:

Explanation:

To consider finding the slope, let's discuss the topic of the gradient.

For a function , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

It is essentially the slope of a multi-dimensional function at any given point

Knowledge of the following derivative rule will be necessary:

Trigonometric derivative: 

Note that u may represent large functions, and not just individual variables!

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

Take the partial derivatives of  at the point 

x:

y:

The slope is 

Example Question #363 : Equations

Find the slope of the function  at the point .

Possible Answers:

Correct answer:

Explanation:

To consider finding the slope, let's discuss the topic of the gradient.

For a function , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

It is essentially the slope of a multi-dimensional function at any given point

Knowledge of the following derivative rule will be necessary:

Derivative of an exponential: 

Note that u may represent large functions, and not just individual variables!

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

 Take the partial derivatives of  at the point 

x:

y:

The slope is 

 

Example Question #362 : Equations

Find the slope of the function  at the point .

Possible Answers:

Correct answer:

Explanation:

To consider finding the slope, let's discuss the topic of the gradient.

For a function , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

It is essentially the slope of a multi-dimensional function at any given point

Knowledge of the following derivative rule will be necessary:

Derivative of an exponential: 

Note that u may represent large functions, and not just individual variables!

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

Take the partial derivatives of   at the point 

x:

y:

The slope is .

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